Abstract:

The Wiggins-Holmes extension of the generalized
Melnikov method (GMM) is applied to weakly damped
parametrically excited cross waves with surface tension in
a long rectangular wave channel in order to determine if
these cross waves are chaotic. The Lagrangian density
function for surface waves with surface tension is
simplified by transforming the volume integrals to surface
integrals and by subtracting the zero variation integrals.
The Lagrangian is written in terms of the three generalized
coordinates (or, equivalently the three degrees of freedom)
that are the time-dependent components of the velocity
potential. A generalized dissipation function is assumed to
be proportional to the Stokes material derivative of the
free surface. The generalized momenta are calculated from
the Lagrangian and the Hamiltonian is determined from a
Legendre transformation of the Lagrangian. The first order
ordinary differential equations derived from the
Hamiltonian are usually suitable for the application of the
GMM. However, the cross wave equations of motion must be
transformed in order to obtain a suspended system for the
application of the GMM. Only three canonical
transformations that preserve the dynamics of the cross
wave equations of motion are made because of an extension
of the Herglotz algorithm to nonautonomous systems. This
extension includes two distinct types of the generalized
Herglotz algorithm (GHA). The system of nonlinear
nonautonomous evolution equations determined from
Hamilton's equations of motion of the second kind are
averaged in order to obtain an autonomous system. The
unperturbed system is analyzed to determine hyperbolic
saddle points that are connected by heteroclinic orbits
The perturbed Hamiltonian system that includes surface
tension satisfies the KAM nondegeneracy requirements; and
the Melnikov integral is calculated to demonstrate that the
motion is chaotic. For the perturbed dissipative system
with surface tension, the Melnikov integral is identically
zero implying that a higher dimensional GMM is necessary in
order to demonstrate by the GMM that the motion is chaotic.
However, numerical calculations of the largest Liapunov
characteristic exponent demonstrate that the perturbed
dissipative system with surface tension is also chaotic. A
chaos diagram is computed in order to search for possible
regions of the damping parameter and the Floquet parametric
forcing parameter where chaotic motions may exist.